Would you be able to provide more details on why you want to just send a Signing Link instead of having that sent through a DocuSign email? I am sure you could probably figure out the link, which based I what I have seen, is values that are added based on the Account, Envelope, and Attachments, but would this be easy to construct piece by piece, not sure. One option is Powerforms, which is a URL that could be sent via email or embedded to a website potentially avoiding email notifications.
On email customization I would point you towards the Resource Files. Here is information on Branding with links to the different Resource Files at the bottom of this article....https://support.docusign.com/en/guides/ndse-admin-guide-advanced-branding-configuration